Three point charges \(+\mathrm{q},+2 \mathrm{q}\) and \(+\mathrm{Q}\) are placed at the three vertices of an equilateral triangle. If the potential energy of the system of three charges is zero, the value of \(Q\) in terms of \(q\) is

Potential energy due to \(+\mathrm{q}\) and \(+2 \mathrm{q}\),
\(
U_1=\frac{K q(2 q)}{r}
\)
\(\therefore \quad\) Potential energy due to \(+\mathrm{q}\) and \(+\mathrm{Q}\),
\(
\mathrm{U}_2=\frac{\mathrm{KqQ}}{\mathrm{r}}
\)
\(\therefore \quad\) Potential energy due to \(+\mathrm{Q}\) and \(+2 \mathrm{q}\),
\(
\mathrm{U}_3=\frac{\mathrm{KQ}(2 \mathrm{q})}{\mathrm{r}}
\)
Given: Potential energy of system \(=0\)
\(
\begin{aligned}
\therefore \quad & \mathrm{U}_1+\mathrm{U}_2+\mathrm{U}_3=0 \\
& \frac{\mathrm{Kq}(2 \mathrm{q})}{\mathrm{r}}+\frac{\mathrm{KqQ}}{\mathrm{r}}+\frac{\mathrm{KQ}(2 \mathrm{q})}{\mathrm{r}}=0 \\
& 2 \mathrm{q}+\mathrm{Q}+2 \mathrm{Q}=0 \\
& 3 \mathrm{Q}=-2 \mathrm{q} \\
& \mathrm{Q}=\frac{-2}{3} \mathrm{q}
\end{aligned}
\)